If $f(x) =\begin{vmatrix}x +λ&x&x\\x&x +λ&x\\x&x&x +λ\end{vmatrix}$, then $f(3x) − f(x) =$

$6x\, λ^2$

We have,

$f(x) =\begin{vmatrix}x +λ&x&x\\x&x +λ&x\\x&x&x +λ\end{vmatrix}$

$⇒f(x)=\begin{vmatrix}3x +λ&x&x\\3x +λ&x +λ&x\\3x +λ&x&x +λ\end{vmatrix}$  [Applying $C_1→C_1+C_2 +C_3$]

$⇒f(x)=(3x +λ)\begin{vmatrix}1&x&x\\1&x +λ&x\\1&x&x +λ\end{vmatrix}$

$⇒f(x)=(3x +λ)\begin{vmatrix}1&x&x\\0&λ&0\\0&0&λ\end{vmatrix}$  [Applying $R_2 →R_2-R_1, R_3→R_3-R_1$]

$⇒f(x)=(3x +λ)λ^2$

$∴ f(3x)-f(x)=(9x+λ) λ^2-(3x+λ)λ^2 = 6x\, λ^2$.